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Stereographic Imaging Condition

One possibility to remove the artifacts caused by the cross-talk between inconsistent reflection events is to modify the imaging condition to use more than one attribute for matching the source and receiver wavefields. For example, we could use the time and slope to match events in the wavefield, thus distinguishing between unrelated events that occur at the same time (Figure 4).

Figure 4.
Comparison of conventional imaging (a) and stereographic imaging (b).
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A simple way of decomposing the source and receiver wavefields function of local slope at every position and time is by local slant-stacks at coordinates ${ \bf x}$ and $t$ in the four-dimensional source and receiver wavefields. Thus, we can write the total source and receiver wavefields ($u_s$ and $u_r$) as a sum of decomposed wavefields ($w_S$ and $w_R$):

$\displaystyle u_s\left ({ \bf x},t \right)$ $\textstyle =$ $\displaystyle \int\!\! w_S\left ({ \bf x},{ \bf p},t \right) d{ \bf p}$ (5)
$\displaystyle u_r\left ({ \bf x},t \right)$ $\textstyle =$ $\displaystyle \int\!\! w_R\left ({ \bf x},{ \bf p},t \right) d{ \bf p} \;.$ (6)

Here, the three-dimensional vector ${ \bf p}$ represents the local slope function of position and time. Using the wavefields decomposed function of local slope, $w_S$ and $w_R$, we can design a stereographic imaging condition which cross-correlates the wavefields in the decomposed domain, followed by summation over the decomposition variable:
r\left ({ \bf x}\right)=
\!\!\!\int\!\!\int\!\! w_S\left (...
w_R\left ({ \bf x},{ \bf p},t \right) d{ \bf p}dt \;.
\end{displaymath} (7)

Correspondence between the slopes p of the decomposed source and receiver wavefields occurs only in planes dipping with the slope of the imaged reflector at every location in space. Therefore, an approximate measure of the expected reflector slope is required for correct comparison of corresponding reflection data in the decomposed wavefields. The choice of the word ``stereographic'' for this imaging condition is analogous to that made for the velocity estimation method called stereotomography (Billette et al., 2003; Billette and Lambare, 1997) which employs two parameters (time and slope) to constrain traveltime seismic tomography.

For comparison with the stereographic imaging condition 7, the conventional imaging condition can be reformulated using the wavefield notation 5-6 as follows:

r\left ({ \bf x}\right)=
\!\!\!\int\!\! \left [\int\!\! w_...
..._R\left ({ \bf x},{ \bf p},t \right) d{ \bf p} \right] dt \;.
\end{displaymath} (8)

The main difference between imaging conditions 7 and 8 is that in one case we are comparing independent slope components of the wavefields separated from one-another, while in the other case we are comparing a superposition of them, thus not distinguishing between waves propagating in different directions. This situation is analogous to that of reflectivity analysis function of scattering angle at image locations, in contrast with reflectivity analysis function of acquisition offset at the surface. In the first case, waves propagating in different directions are separated from one-another, while in the second case all waves are superposed in the data, thus leading to imaging artifacts (Stolk and Symes, 2004).

Figure 3(b) shows the image produced by stereographic imaging of the data generated for the model depicted in Figures 1(a)-1(b), and Figure 5(b) shows the similar image for the model depicted in Figures 2(a)-2(b). Images 3(b) and 5(b) use the same source receiver wavefields as images 3(a) and 5(a), respectively. In both cases, the cross-talk artifacts have been eliminated by the stereographic imaging condition.

ii kk
Figure 5.
Images obtained for the model in Figures 2(a)-2(c) using the conventional imaging condition (a) and the stereographic imaging condition (b).
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ttr1 ii1 ttr2 ii2 ttr0 ii0 vel kk
Figure 6.
Data corresponding to shots located at coordinates $x=16$ kft (a), $x=24$ kft (c), and the sum of data corresponding to both shot locations (e). Image obtained by conventional imaging condition for the shots located at coordinates $x=16$ kft (b), $x=24$ kft (d) and the sum of data for both shots (f). Velocity model extracted from the Sigsbee 2A model (g) and image from the sum of the shots located at $x=16$ kft and $x=24$ kft obtained using the stereographic imaging condition (h).
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Next: Example Up: Sava: Stereographic imaging Previous: Conventional imaging condition